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Simplifying 2r2 + 10r + 11 = 0 Reorder the terms: 11 + 10r + 2r2 = 0 Solving 11 + 10r + 2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 5.5 + 5r + r2 = 0 Move the constant term to the right: Add '-5.5' to each side of the equation. 5.5 + 5r + -5.5 + r2 = 0 + -5.5 Reorder the terms: 5.5 + -5.5 + 5r + r2 = 0 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + 5r + r2 = 0 + -5.5 5r + r2 = 0 + -5.5 Combine like terms: 0 + -5.5 = -5.5 5r + r2 = -5.5 The r term is 5r. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5r + 6.25 + r2 = -5.5 + 6.25 Reorder the terms: 6.25 + 5r + r2 = -5.5 + 6.25 Combine like terms: -5.5 + 6.25 = 0.75 6.25 + 5r + r2 = 0.75 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (r + 2.5) equal to 0.866025404 and -0.866025404.Subproblem 1
r + 2.5 = 0.866025404 Simplifying r + 2.5 = 0.866025404 Reorder the terms: 2.5 + r = 0.866025404 Solving 2.5 + r = 0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 0.866025404 + -2.5 r = 0.866025404 + -2.5 Combine like terms: 0.866025404 + -2.5 = -1.633974596 r = -1.633974596 Simplifying r = -1.633974596Subproblem 2
r + 2.5 = -0.866025404 Simplifying r + 2.5 = -0.866025404 Reorder the terms: 2.5 + r = -0.866025404 Solving 2.5 + r = -0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -0.866025404 + -2.5 r = -0.866025404 + -2.5 Combine like terms: -0.866025404 + -2.5 = -3.366025404 r = -3.366025404 Simplifying r = -3.366025404Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.633974596, -3.366025404}
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